Congruent



|Day |In Class |Homework |Completed |

|Day 1 |Classifying Triangles | | |

| | |D1 HW – pg. 5 | |

|Day 2 |Angles of Triangles | | |

| | |D2 HW – pg. 11 | |

|Day 3 | | | |

| |Angles of Triangles and Congruent |Skills Practice 4-2 | |

| |Triangles | | |

|Day 4 | | D4 HW – pg. 17 | |

| |Isosceles and Equilateral Triangles | | |

|Day 5 |Coordinate Triangle Proofs |D5 HW – pg. 22 (#1) | |

|Day 6 |Coordinate Triangle Proofs |D6 HW – pg. 22 (#2) | |

|Day 7 |Congruent Triangles |D7 HW – pg. 26 | |

|Day 8 |SSS and SAS |Worksheet in Packet | |

|Day 9 |ASA and AAS |Review Packet | |

|Day 10 |Review |Study | |

|Day 11 |Test |Good Luck! | |

Classifying Triangles

Review: Given the triangle below identify the following:

• The sides of [pic] are _____, _____, and _____.

• The vertices of [pic] are _____, _____, and _____.

• The angles of [pic] are _____, _____, and _____.

Classifying Triangles by ANGLES

|Acute Triangle |Right Triangle |Obtuse Triangle |Equiangular Triangle |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

|all acute angles |one right angle |one obtuse angle |all equal (congruent) angles |

*** There can be at most one right or obtuse angles in a triangle.

➢ If all three angles of an acute triangle are congruent, then the triangle is an _______________ triangle.

➢ If one of the angles of a triangle is a right angle, then the triangle is a _______________ triangle.

➢ If all three angles of a triangle are acute, then the triangle is an _______________ triangle.

➢ If one of the angles of a triangle is an obtuse angle, then the triangle is an _______________ triangle.

Classify each of the triangles below as acute, equiangular, obtuse or right.

1. 2.

3. 4.

5. Classify each triangle below as acute, equiangular, obtuse or right.

(PQS is a(n) _______________ triangle.

(QRS is a(n) _______________ triangle.

(PQR is a(n) _______________ triangle.

6. Classify each triangle below as acute, equiangular, obtuse or right.

(BAD is a(n) _______________ triangle.

(BCD is a(n) _______________ triangle.

(ABC is a(n) _______________ triangle.

Classifying Triangles by SIDES

|Equilateral Triangle |Isosceles Triangle |Scalene Triangle |

| | | |

| | | |

| | | |

| | | |

| | | |

| |two sides congruent |no sides congruent |

|all sides congruent | | |

➢ If two sides of a triangle are congruent, then the triangle is an _______________ triangle.

➢ If no sides of a triangle are congruent, then the triangle is a _______________ triangle.

➢ If all three sides of a triangle are congruent, then the triangle is an _______________ triangle.

1. If point M is the midpoint of JL, classify each triangle as equilateral, isosceles, or scalene.

(JKM is a(n) _______________ triangle.

(KML is a(n) _______________ triangle.

(JKL is a(n) _______________ triangle.

2. Classify each triangle as equilateral, isosceles or scalene.

a. b.

3. Find the measures of the sides of isosceles triangle ABC.

• AB = _____

• AC = _____

• BC = _____

4. Find the measures of the sides of equilateral triangle FGH.

5. Find the value of x if MN ( LN

Day 1 HW

1. Classify each triangle as acute, equiangular, obtuse or right.

a. (UYZ

b. (UXZ

c. (UWZ

d. (UXY

2. C is the midpoint of BD and E is the midpoint of DF. Classify each triangle as equilateral, isosceles or scalene.

a. (ABC

b. (ADF

c. (ABD

3. (ABC is an isosceles triangle with AB ( BC. Find x and the measure of each side.

4. (FGH is an equilateral triangle. Find x and the measure of each side.

5. Classify each triangle by its angles and sides.

a. (ABE

b. (EBC

c. (BDC

Angle Relationships in Triangles

1. Find m(ABC

2. Find the measures of each numbered angle.

(1 = _________

(2 = _________

(3 = _________

(4 = _________ (7 = _________

(5 = _________ (8 = _________

(6 = _________ (9 = _________

3. The measures of the angles of a triangle are in the extended ratio 8:6:2. Find the measure of the largest angle.

4. The measures of the angles of a triangle are in the extended ratio 4:5:11. Find the measure of the smallest angle.

5. Find the value of each angle.

The corollary below (a theorem with a proof that follows as a direct result of another theorem) follows directly from the Triangle Sum Theorem.

1. Find m(C.

An exterior angle of a triangle is formed by one side of the triangle and the extension of an adjacent side.

m(1 + m(2 = m(4

Example:

60 + x = 111

x = 51

1. Find m(1.

2. Find x.

a. b.

3. Find m(D.

4. Find m(JKL.

5. Find m(PRS.

6. Find the measures of each numbered angle.

(1 = _________

(2 = _________

(3 = _________

(4 = _________

Practice:

Find the measure of each unknown angle.

1. 2.

3. 4.

5.

Day 2 HW

1. Find the measure of each numbered angle.

a. b.

2. Find the value of x.

a. b.

3. Find the measure of each angle.

a. b.

c. d.

If m(2 ( m(5 and m(3 ( m(6, then

m(1 ( m(4

The graphic organizer describes the relationships of interior and exterior angles in a triangle. Use the word bank to correctly identify each.

Third Angle Theorem Right Triangles Exterior Angle Theorem

Triangle Sum Theorem Equiangular Triangles

Isosceles and Equilateral Triangles

In an isosceles triangle,________ sides are equal, therefore ________ angles are equal.

Vertex Angle-

Base-

Base Angles-

Legs-

|Theorem |Example |

|Isosceles Triangle Theorem | |

|If two sides of a triangle are congruent, then the angles opposite the|If RT ( RS, then |

|sides are congruent. |(_____ ( (_____. |

| | |

| | |

|Converse of Isosceles Triangle Theorem (ITBA) | If (N ( (M, then |

|If two angles of a triangle are congruent, then the sides opposite |_____ ( _____. |

|those angles are congruent. | |

Remember Equilateral Triangles:

➢ all sides are ________ to each other.

RT ( RS ( ST

➢ all angles are ________.

(R ( (S ( (T

1. Find the m(B and m(C if m(A = 53.

2. Find the measure of each angle.

a. b.

m(C = ________ m(P = ________

c. d.

m(H = ________ m(A = ________ m(B = ________

3. Given the isosceles triangle, find m(Y and m(Z.

4. (BCA is an equilateral triangle. Find the value of x and y.

5. Find the value of x.

6. Find the measure of the vertex angle if each base angle is 80.

7. Find the measure of each base angle if the vertex angle is 76.

8. What is the measure of degrees in each acute angle of an isosceles right triangle?

9. The degree measure of the vertex angle A of an isosceles triangle is 110. Find the other angles.

10. In the accompanying diagram, triangle ABC and triangle ABD are isosceles triangles with the m(CAB = 50 and m(BDA = 55. If AC = AB and AB = BD, what is the m(CBD?

11. The measure of the vertex angle of an isosceles triangle is 15 more than each base angle. Find the number of degrees in each angle.

12. The measure of the base angle of an isosceles triangle is seven times the measure of the vertex angle. Find the number of degrees in each angle.

13. In isosceles triangle ABC, the measure of the vertex angle C is 30 more than each base angle. Find the number of degrees in each angle.

14. The measure of the exterior angle to a base angle of an isosceles triangle is 115. What is the measure of the vertex angle of the triangle?

15. In a triangle, the measure of the second angle is 30 more than the measure of the first angle. The measure of the third angle is 45 more than the first angle. Find the number of degrees in each angle of the triangle. What type of triangle is it?

16. In triangle ABC, m(A = x, m(B = x + 10, and the measure of an exterior angle at C is 70. Find the value of x.

Day 4 HW

1. Find each measure.

a. m(BAC b. m(SRT

c. CB d. TR

2. Find the value of x.

a. b. c.

3. Find each measure.

a. m(CAD

b. m(ACD

c. m(ACB

d. m(ABC

Triangles and Coordinates Proofs

Coordinate proofs use figures in the coordinate plane and algebra to prove geometric concepts.

To Do Coordinate Geometry Proofs:

1. Graph the figure.

2. Use one or a combination of the following formulas:

➢ Distance formula to show that __________ are __________.

➢ Slope formula to show that __________ are __________ or have ____________________.

➢ Midpoint formula to show that __________ have the __________ midpoint.

3 Show ALL work.

4. Write a statement(s) to explain why it is that figure to finish the proof.

Properties of an Isosceles Triangle:

1. An isosceles triangle has __________ sides __________.

2. An isosceles triangle has __________ angles __________.

Show two sides have the same distance. That means sides will have the same __________.

➢ To prove that 2 sides have the same length, use the __________ formula. You will do the distance formula _____ times.

Statement(s):

1. The coordinates of triangle ABC are A(5, 4), B(8, 1), and C(2, 1). Prove that ABC is an isosceles triangle. Hint: Show that AB ( AC.

2. The coordinates of triangle ABC are A(3, 1), B(1, -1), and C(5, -1). Prove that ABC is an isosceles triangle.

Properties of a Right Triangle:

1. A right triangle has _____ right __________.

Show the triangle has a right angle. That means the slopes of the legs of the triangle will have ____________________________.

➢ To prove that there is a right angle, use the __________ formula. You will do the slope formula _____ times. The slopes will have ____________________.

Statement(s):

1. The coordinates of triangle ABC are A(-1, 1), B(-4, 1), and C(-1, 3). Prove that ABC is a right triangle. Hint: Show that AB and AC form a right angle.

2. The coordinates of triangle ABC are A(-1, -1), B(2, -3), and C(-1, -3). Prove that ABC is a right triangle.

Properties of an Isosceles Right Triangle:

1. An isosceles right triangle has _____ right __________.

2. An isosceles right triangle has _____ equal __________.

Show two sides have the same distance and there is a right angle.

➢ To prove a triangle is isosceles, use the __________ formula. You will do the distance formula _____ times. The lengths will have ____________________.

AND

➢ To prove that there is a right angle, use the __________ formula. You will do the slope formula _____ times. The slopes will have ____________________.

Statement(s):

1. The coordinates of triangle RST are R(0, 1), S(4, 5), and T(4, 1). Prove that RST is an isosceles right triangle.

Days 5 and Day 6 Homework

1. The coordinates of triangle ABC are A(2, 5), B(5, 2), and C(-1, 2). Prove that ABC is an isosceles triangle.

2. The coordinates of triangle ABC are A(-1, 1), B(1, -2), and C(-1, -2). Prove that ABC is a right triangle.

Congruent Triangles

Remember: congruent means ________ shape, ________ size. The symbol for congruent is _____.

Triangles are congruent if they have the same size and shape. Their corresponding parts, the angles and sides that are in the same position, are congruent.

|Corresponding Parts |

|Congruent Angles |Congruent Sides |

|(A ( (J |[pic] ( [pic] |

|(B ( (L |[pic] ( [pic] |

|(C ( (K |[pic] ( [pic] |

To identify corresponding parts of congruent triangles, look at the order of the vertices in the congruence statement such as (ABC ( (JLK.

1. Given: (XYZ ( (NPQ. Identify the congruent corresponding parts.

a. (Q ( _____ b. [pic] ( _____ c. (P ( _____

d. (X ( _____ e. [pic] ( _____ f. [pic] ( _____

g. Write a congruence statement.

2. Use the given information to find the measures of the angles.

(TPR is equiangular.

a. m(QRP = ______

b. m(TRP = ______

c. m(RTS = ______

d. m(TRS = ______

3. Use the figure to find the following angles.

a. m(A = ______

b. m(B = ______

c. m(BCF = ______

d. m(EFD = ______

4. Find the value of the x.

In two congruent polygons, all of the parts of one polygon are congruent to the corresponding parts or matching parts of the other polygon.

Use tick marks and arcs to identify corresponding angles and corresponding sides.

Write a congruence statement.

Congruence Statement: ___________________________________________

Polygon BCDE ( polygon RSTU. Find each value.

a. x

b. y

c. w

d. z

Like congruence of segments and angles, congruence of triangles is reflexive, symmetric, and transitive.

❖ Reflexive Property of Triangle Congruence:

Ex: (A ( (A , AB ( AB

(ABC ( _____

❖ Symmetric Property of Triangle Congruence:

Ex: AB ( BA

If (ABC ( (EFG, then _______________

❖ Transitive Property of Triangle Congruence:

Ex: If (1 ( (2 and (2 ( (3, then (1 ( (3

If (ABC ( (EFG and (EFG ( (JKL , then _______________

Day 7 HW

1. Show that the triangles are congruent by identifying all congruent parts. Then write a congruence statement.

a. (R ( _____ b. [pic] ( _____ c. (S ( _____

d. (T ( _____ e. [pic] ( _____ f. [pic] ( _____

g. Congruence Statement _____________________

2. Find x and y.

3. Find each measure.

a. m(1

b. m(2

c. m(3

4. Which is a factor of x2 + 19x – 42?

a. x + 14 b. x + 2 c. x – 2 d. x – 14

5. Find the distance between points (5, 7) and (-2, 3).

Proving Triangles Congruent

To prove triangles are congruent, you must find _____ corresponding parts that match up.

Triangle Congruence Theorems:

▪ SSS Congruence

When proving triangles congruent with SSS, you must find _____ ( sides.

You write in Justifications: _______________.

▪ SAS Congruence

When proving triangles congruent with SAS, you must find _____ ( sides and _____ ( angle. You must have _____ included ___________.

The word included means __________.

You write in Justifications: _______________.

▪ ASA Congruence

When proving triangles congruent with ASA, you must find _____ ( side and _____ ( angles. You must have _____ included ___________.

You write in Justifications: _______________.

▪ AAS Congruence

When proving triangles congruent with AAS, you must find _____ ( side and _____ ( angles. The side is not included.

You write in Justifications: _______________.

From the given information, what theorem (SSS, SAS, ASA or AAS) would prove the triangles congruent.

1. (A ( (C, AD ( DC 2. HE ( FG, (EFH ( (FHG, (EHF ( (GFH

3. (U ( (W, UV ( WV, UX ( WX 4. UV ( WV, UX ( WX

5. (Y ( (B, YA ( BA 6. (DEG ( (FEG, DE ( EF

7. (HJK ( (MLK, JH ( ML 8. (A ( (D, AB ( DE, (F ( (C

Write a two-column proof.

1. Given: AB ( DB

C is the midpoint of AD.

Prove: (ABC ( (DBC

2. Given: AB ll CD

(CAB ( (ACD

Prove: (ACD ( (ABC

3. Given: RS ( TU, RT ( US

Prove: (RST ( (UTS

4. Given: (S ( (V

T is the midpoint of SV

Prove: (RST ( (UTV

5. Given: CD bisects AB

AC ( BC

Prove: [pic]ADC ( [pic]BDC

6. Given: T is the midpoint of PQ

PQ bisects RS

RQ ( SP

Prove: [pic]RTQ ( [pic]STP

7. Given: [pic]

[pic]

Prove: [pic]

8. Given: [pic]bisect each other at E

Prove: [pic]

|Day |In Class |Homework |

|Day 1 |Congruent Triangles |Pgs. 257-261 #13-16, 18, 28, 46, 48-50 |

|Day 2 |SSS and SAS |Pgs. 268-270 #16-19, 27, 28, 35, 46-48 |

|Day 3 |ASA and AAS |Worksheet in Packet |

Extra DO NOT COPY

1. 2. 3.

4. 5.

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B

A

Statements

Reasons

50

58

72

53

31

37

45

104

60

60

60

30

40

70

49

34

97

60

70

60

60

60

C

A

S

B

R

Q

J

P

12

M

4

4

11

.75

B

9

1.3

7

8

8

L

4

K

1.5

C

9x - 1

5x – 0.5

4x + 1

5

A

30

60

60

60

30

120

B

C

D

B

A

8.4

2y + 5

3y - 3

5y - 19

G

H

F

17

3x - 4

2x + 7

L

N

M

Triangle Sum Theorem - all angles of a triangle add up to 180(.

m(1 + m(2 + m(3 = 180

45

63

72

72 + 63 + 45 = 180

A

C

64

50

K

x

L

J

57

2

1

28

M

3x

71

3

4

W

5

6

7

67

X

9

65

8

58

V

Y

Z

Exterior Angle Theorem – the sum of the two nonadjacent (remote) interior angle = exterior angle

m(1 + m(2 = m(4

4

3

2

1

exterior angle

nonadjacent(remote) interior angles

50

2x - 15

J

W

88

55

x

40

x

L

K

x

60

x

exterior angle

111

nonadjacent(remote) interior angles

80

1

60

68

D

B

4x + 5

C

A

S

Q

5x - 1

9x + 2

P

R

23

The acute angles of a right triangle are complementary.

m(1 + m(2 = 90

2

1

B

39

A

C

R

T

7x - 13

4x + 9

2x + 2

S

T

X

x

Z

1

2

Y

4

52

3

38

45

56

102

5

2

4

6

53

112

32

x

x

17

z

y

x

29

y

z

22

31

3

2

Third Angle Theorem – If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent.

6

1

4

E

B

C

5

D

F

A

L

M

J

K

H

Z

A

Y

B

C

E

D

F

G

X

U

W

V

X

U

W

V

D

C

A

B

G

Statements

H

E

F

How are they different?

How are they alike?

AAS

ASA

How are they different?

How are they alike?

SAS

SSS

R

T

D

S

U

C

B

A

40

40

x

B

80

3x + 10

2x + 20

A

F

D

E

C

Q

30

T

P

S

R

X

Z

Y

P

Q

N

C

B

K

J

A

L

Two tick marks shows these sides are congruent.

A

Three arcs shows these angles are congruent.

A

B

C

T

R

S

L

M

N

B

C

A

R

S

T

X

Z

Y

P

8

8

40

2x

3

4y - 5

C

A

B

2x

62

8

8

Y

X

W

4x - 2

R

78

Reasons

Q

C

A

B

86

8x

J

G

H

6x + 18

78

C

A

B

C

D

S

U

V

T

S

R

Reasons

Statements

A

B

C

D

X

U

V

T

S

R

S

D

A

B

C

Q

P

R

T

A

C

D

B

Reasons

Statements

T

R

U

R

Reasons

Statements

Reasons

Statements

Reasons

Statements

D

B

C

E

A

Reasons

Statements

4

35

R

36

2

3

1

80

U

T

V

S

A

D

B

C

47

43

60

4

20

120

U

W

Z

Y

60

60

10

5

A

F

E

15

B

C

D

8

9x - 8

3x + 10

6x + 1

A

B

C

x - 3

4x - 21

2x - 7

G

H

F

75

15

75

A

B

C

D

52

15

E

1

4x

100

x - 5

2x - 15

148

3

22

43

27

2x + 27

2x - 11

31

3

24

23

W

Z

X

Y

2

1

105

2x

22

3

1

2

x

24

2x

3x

R

P

T

4

4

C

B

A

60

50

T

S

R

6x - 9

2x + 11

3

55

55

C

B

A

3x + 6

D

x2 + 5x

C

B

A

92

E

C

D

U

4w - 7

z + 16

R

S

T

B

2w + 13

3z + 10

11

12

2y - 31

2x + 9

y + 11

49

T

R

J

L

S

y

K

40

2x

3

74

15

2

1

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